A solution curve of the differential equation | vedclass

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(D) The given differential equation is $(x+2)(x+2+y) \frac{dy}{dx}-y^2=0$.Substitute $y=(x+2)t$,then $\frac{dy}{dx}=(x+2) \frac{dt}{dx}+t$.Substitutin

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$A, D$

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(D) The given differential equation is $(x+2)(x+2+y) \frac{dy}{dx}-y^2=0$.Substitute $y=(x+2)t$,then $\frac{dy}{dx}=(x+2) \frac{dt}{dx}+t$.Substituting this into the equation: $(x+2)(x+2+(x+2)t) \frac{dy}{dx}-((x+2)t)^2=0$.$(x+2)^2(1+t) \frac{dy}{dx}-(x+2)^2t^2=0$.$(1+t)((x+2) \frac{dt}{dx}+t)-t^2=0$.$(1+t)(x+2) \frac{dt}{dx}+t+t^2-t^2=0$.$(1+t)(x+2) \frac{dt}{dx}+t=0$.Separating variables: $\frac{1+t}{t} dt = -\frac{dx}{x+2}$.Integrating both sides: $\int (\frac{1}{t}+1) dt = -\int \frac{dx}{x+2}$.$\ln|t|+t = -\ln|x+2|+C$.Substituting $t=\frac{y}{x+2}$: $\ln(\frac{y}{x+2})+\frac{y}{x+2} = -\ln(x+2)+C$.$\ln y - \ln(x+2) + \frac{y}{x+2} = -\ln(x+2)+C$.$\ln y + \frac{y}{x+2} = C$.Since the curve passes through $(1,3)$,$\ln 3 + \frac{3}{1+2} = C \Rightarrow \ln 3 + 1 = C$.The solution curve is $\ln y + \frac{y}{x+2} = \ln 3 + 1$.For option $(A)$,intersect $y=x+2$: $\ln(x+2) + \frac{x+2}{x+2} = \ln 3 + 1 \Rightarrow \ln(x+2) + 1 = \ln 3 + 1 \Rightarrow x+2=3 \Rightarrow x=1$. Only one point.For option $(D)$,$y=(x+3)^2$: $\ln(x+3)^2 + \frac{(x+3)^2}{x+2} = \ln 3 + 1$. Let $f(x) = 2\ln(x+3) + \frac{(x+3)^2}{x+2} - \ln 3 - 1$. Since $f(x)$ is strictly increasing for $x>0$ and $f(0) > 0$,it does not intersect.

Column $I$	Column $II$
$(A)$ The number of solutions of the equation $x e^{\sin x}-\cos x=0$ in the interval $(0, \frac{\pi}{2})$	$(p)$ $1$
$(B)$ Value$(s)$ of $k$ for which the planes $k x+4 y+z=0, 4 x+k y+2 z=0$ and $2 x+2 y+z=0$ intersect in a straight line	$(q)$ $2$
$(C)$ Value$(s)$ of $k$ for which $\|x-1\|+\|x-2\|+\|x+1\|+\|x+2\|=4 k$ has integer solution$(s)$	$(r)$ $3$
$(D)$ If $y^{\prime}=y+1$ and $y(0)=1$,then value$(s)$ of $y(\ln 2)$	$(s)$ $4$
	$(t)$ $5$

$A$ solution curve of the differential equation $(x^2+xy+4x+2y+4) \frac{dy}{dx}-y^2=0, x>0$,passes through the point $(1,3)$. Then the solution curve

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